Dtjytyk

It is actually impossible to have a unique and unambiguous answer to the puzzle without explicitly articulating a probability model for how the information on gender and birthday is generated. The reason is that (1) for the problem to have a unique answer some random process is required, and (2) the answer is a function of which random model is used.

1. The problem assumes that a unique probability can be deduced as the answer. This requires that the set of children described is chosen by a random process, otherwise the number of boys is a deterministic quantity and the probability would be 0 or 1 but with no ability to determine which is the case. More generally one can consider random processes that produce the complete set of information referenced in the problem: choose a parent, then choose what to reveal about the number, gender, and birth days of its children.




2. The answer depends on which random process is used. If the Tuesday birth is disclosed only when there are two boys, the probability of two boys is 1. If Tuesday birth is disclosed only when there is a sister, the probability of two boys is 0. The answer could be any number between 0 or 1 depending on what process is assumed to produce the data.

There is also a linguistic question of how to interpret "one is a boy born on Tuesday". It could mean that the number of Tuesday-born males is exactly one, or at least one child.

I guess the following two versions of framing the question yield two different probabilities:

1. Dave has two children. Is atleast one of them a boy who is born on Tuesday? Dave answers Yes.




2. Dave has two children. I ask him to first choose and fix one child at random, and tell me if it is a boy who was born on Tuesday. Dave answers yes he is a boy born on Tuesday.

For 1st the probability (of both being boys) is 13/27, while for the second the probability is 1/2.

The way in which the question is asked, it's in line with 1st, hence the answer should be 13/27.

There is always room for misinterpreting a question when one does not fully understand the language in which it is written. I think that the way mathematics and mathematicians use conditional probability is clear:

$$P(A|B)=P(A cap B)/P(B).$$

So I believe that this is the interpretation that one should take, and thus arrive at your answer of 13/27, and not search for further nuances, which are not too difficult to find.

Well, given the unstated assumption that the writer is a mathematician and therefore not using regular english, then I agree with the 13/27 answer.

But in everyday english, from "there are two fleems, one is a glarp" we all infer that the other is not a glarp.

From "there are two fleems, one is a glarp, which is snibble" we would still infer that the other is not a glarp. Whereas from "there are two fleems, one is a glarp which is snibble" (absence of comma, or when spoken, difference in intonation) we would infer that the other is not a snibble glarp, but it could still be an unsnibble glarp.

The Tuesday is a red herring. It's stated as a fact, thus the probability is 1. Also, it doesn't say "only one boy is born on a Tuesday". But indeed, this could be a language thing.

With 2 children you have the following possible combinations:

1. two girls

2. a boy and a girl

3. a girl and a boy

4. two boys

If at least 1 is a boy we only have to consider the last three combinations. That gives us one in three that both are boys.

The error which is often made is to consider 2. and 3. as a single combination.

edit

I find it completely counter-intuitive that the outcome is influenced by the day, and I simulated the problem for one million families with 2 kids. And lo and behold, the outcome is 12.99 in 27. I was wrong.

This, in my opinion, is why the intuitive approach fails:

One has a tendency to think that
the probability of 7*P(b AND d1) = P(b AND d1) + P(b AND d2) + ... + P(b AND d7) = P((b AND d1) OR (b AND d2) OR ... OR (b AND d7)) = P(b AND (d1 OR d2 OR ... OR d7)) = P(b).

However, the flaw here is that, in reality, P(b AND d1) + P(b AND d2) + ... + P(b AND d7) is NOT equal to P((b AND d1) OR (b AND d2) OR ... OR (b AND d7)). This means that mentioning independent (and one might think irrelevant) information alongside with relevant information actually changes the resulting probabilities.

One interesting consequence: if I say something like
"I have two children. One of them is a boy who was born at 10:24 PM on February 10th,"
The probability that I have two boys is now almost exactly the same as as the probability that I have a girl and a boy. Adding a unique or almost unique piece of information makes the stuff I want to know about the other child independent of the information I have on the first child. If I took this to the extreme and said that I have a firstborn boy, won't know anything additional about the other child.

The problem lacks a scenario, "situation" or "experiment" [1], according to probability theory, and is therefore, as it stands, unsolvable.

However, assuming a general scenario supporting the problem description: walking down the street, I met a man claiming to have two children of which one boy born on a Tuesday. Is it more likely the other child is a boy, or the same? the correct answer is it is equally likely.

Notice how the problem now does not ask for exact probabilities, but rather which of two probabilities is higher. This is better because a lot of the man's actions can now be disregarded: his lying for example.

Assuming equal birth rates among boys and girls: a quick explanation of the answer: the man is just as likely to claim the sex and birth day of the other child. The situation is therefore symmetric, and it is really nothing special about the claim. The probability space can thusly be partitioned into two symmetric parts of equal probability: one where a claim is made about the firstborn, and the other about the second born. Since any such claim is made irrespective of the sex of the other child, its sex must be of equal probability. And since the two parts have equal probability, the sex of the other child has equal probability.

For a more thorough explanation, add a request in the comment section and I will update the answer.

[1] Wikipedia on "probability space"

The VERY simple answer is that you can get rid of all ambiguity if you can clarify from what pool the parent was chosen. This allows you to restate the "probability" question as a "percentage" question, taking idealized percentages.

Consider:

Given a parent randomly selected from the pool of all parents who have two children where at least one of the children is a boy born on Tuesday, what is the probability that both children are boys?

This can be stated as a percentage question:

What percentage of (parents who have two children where at least one of the children is a boy born on Tuesday) have two boys?

By contrast, if the day was a matter of chance (not a restriction on the pool size), we get a different question:

Given a parent randomly selected from the pool of all parents who have two children at least one of whom is a boy, where after the random selection is made we shall be told the day of the week on which the boy was born (if there is only one boy) or if there are two boys we shall be told the day of the week on which a randomly chosen one of the two boys was born, what is the probability that both children are boys if the day we are told is Tuesday?

As you can see, in this setup the "Tuesday" part has absolutely no influence on the selection process, and can be entirely disregarded.

This is what is meant by Wikipedia's statement:

The moral of the story is that these probabilities do not just depend on the known information, but on how that information was obtained.

As a further note, you can't "prove" anything about the "actual" meaning by using computer simulations, because in order to program a computer simulation in the first place you must first disambiguate which scenario you are actually talking about. So the only thing a computer simulation can "prove" is how the programmer interpreted the question.